I am given a field $F=\mathbb{Q}$ and two sets of vectors:
(i) $v_1 = (1 0 1 1)$ $v_2 = (1001)$ $v_3=(1101)$ $v_4=(1111)$
(ii) $v_1=(111-1)$ $v_2=(1110)$ $v_3=(0110)$ $v_4=(0010)$
and the question asks whether changing the field will affect the fact of whether vectors are linearly dependent or independent.
For the first set I got that they are linearly dependent and for the second - linearly independent. I tried changing the field $F=\mathbb{Q}$ to $F=\mathbb{F}_2$ and seeing what happens and it seems that the result did not change. So, I guess changing the field will not affect linear dependence/independence in these cases but I am not sure what the reasons are for that and how to explain it properly instead of using examples.
Thank you in advance!
This depends on what kind of "change of field" you are talking about.
1) If you have vectors from the space $\mathbb{F}^n$, and then go to a field extension $\mathbb{F'}$ such that $\mathbb{F}\subset\mathbb{F'}$, then the the linear dependence does not change. You can see this by realizing that you can express linear dependence using determinants, whose value does not depend on a field extension. An example of this are vectors over $\mathbb{Q}$. It does not matter if you go to real number, or complex, or algebraic, or anything like that.
2) But on the other hand if you just interpred the expression $(2,2,2)$ over different field, the answer is yes, it depends. Because (as @Morgan already mentioned), in $\mathbb{F}_2$, this is the zero vector. In most other fields its not. But i would say these are not really the same vectors in the beginning, just the notation happens to coincide. So in this sense, the question might not really be well posed.